quan

t-ph

/950

3009

12

Mar

199

5

quant-ph/9503009THEP-95-2March 1995

Standard Model plus Gravityfrom Octonion Creators and Annihilators

Frank D. (Tony) Smith, Jr.e-mail: gt0109e@prism.gatech.edu

and fsmith@pinet.aip.orgP. O. Box for snail-mail:

P. O. Box 430, Cartersville, Georgia 30120 USAWWW URL http://www.gatech.edu/tsmith/home.html

School of PhysicsGeorgia Institute of Technology

Atlanta, Georgia 30332

Abstract

Octonion creation and annihilation operators are used to constructthe Standard Model plus Gravity. The resulting phenomenologicalmodel is the D4 −D5 − E6 model described in hep-ph/9501252.

c©1995 Frank D. (Tony) Smith, Jr., Atlanta, Georgia USA

Contents

1 Introduction. 2

2 Octonion Creators and Annihilators. 3

3 Complexified Octonions. 10

4 Dimensional Reduction. 11

5 Spacetime and Internal Symmetries. 145.1 Spacetime, Gravity, and Phase . . . . . . . . . . . . . . . . . . 145.2 Internal Space and Symmetries. . . . . . . . . . . . . . . . . . 15

1

1 Introduction.

The purpose of this paper is to outline a way to build a model of the StandardModel plus Gravity from the Heisenberg algebra of fermion creators andannihilators.

We want to require that the superposition space of charged fermion cre-ation operators be represented by multiplication on a continuous unit spherein a division algebra. That limits us to:

the complex numbers C, with parallelizable S1,

the quaternions Q, with parallelizable S3, and

the octonions O, with parallelizable S7.

We choose the octonions because they are big enough to make a realisticphysics model.

Octonions are described in Geoffrey Dixon’s book [3] and subsequent pa-pers [4, 5, 6, 7], and in Ian Porteous’s book [11]. Essential mathematicaltools include the octonion X-product of Martin Cederwall [2] and the octo-nion XY-product of Geoffrey Dixon [7].

The purpose of this paper is to build a physics model, not to do math-ematics, so I ignore mathematical details and subtle points. For them, seethe references.

This paper is the result of discussions with Ioannis Raptis and SarahFlynn, and reading a preprint of Steve Selesnick on fermion creation operatorsas fundamental to the Quantum Net of David Finkelstein. John Caputlu-Wilson has discussed the role of propagator phase. Igor Kulikov and TangZhong have also discussed the paper, and Igor has made it clear that I shouldnot misspell Shilov as Silov.

2

2 Octonion Creators and Annihilators.

Consider the octonions O and their unit sphere S7.

Our starting point is the creation operator αOL for the first generationoctonion fermion particles. In the octonion case, the L denotes only thehelicity of the neutrino, which is a Weyl fermion. The other fermions areDirac fermions, and can exist in either helicity state L or R.

If a basis for the octonions is {1, e1, e2, e3, e4, e5, e6, e7, }, then the firstgeneration fermion particles are represented by:

Octonion Fermion Particlebasis element

1 e− neutrinoe1 red up quarke2 green up quarke6 blue up quarke4 electrone3 red down quarke5 green down quarke7 blue down quark

(1)

Graphically, represent the neutral left-handed Weyl e-neutrino creationoperator ανeL by

r -ανeL

Now, represent the charged left-handed and right-handed Dirac electron-

3

quark creation operators αeqL and αeqL by vectors to a point on the sphereS7 (represented graphically by a circle):

����r 6

?

-

αeqL

αeqR

ανeL

Any superposition of charged fermion particle creation operators αeqL andαeqR can be represented as a point on the sphere S7 defined by their repre-sentative vectors. The sphere S7 should be thought of as being orthogonalto the vector ανeL.

We will represent the superposition of creation of e-neutrinos (representedby a vector on a line) and charged particles (represented by vectors to asphere S7) by letting the magnitude of the amplitude | ανeL | of the e-neutrino creator vector run from 0 to 1 and then determining the radius r ofthe sphere S7 in octonion space O by

| ανeL |2 +r2 = 1 (2)

We now have as representation space for the octonion creation operatorsS7 × RP 1, where we have parameterized RP 1 by the interval [0, 1) ratherthan the conventional [0, π).

The octonion first-generation fermion annihilation operator, or antiparti-cle creation operator, is α†OR.

Therefore, for the octonions, we have the nilpotent Heisenberg algebramatrix:

0 αOL β

0 0 α†OR

0 0 0

(3)

4

How does this correspond to the D4 −D5 − E6 model described in hep-ph/9501252 [13]?

The octonion fermion creators and annihilators,

0 αOL 0

0 0 α†OR

0 0 0

(4)

are both together represented in the D4 −D5 − E6 model by the Shilovboundary of the bounded complex homogeneous domain corresponding tothe Hermitian symmetric space E6/(D5 × U(1)).(A good reference on Shilov boundaries is Helgason [10].)The Shilov boundary is two copies of S7 ×RP 1. The RP 1 part representsthe Weyl neutrino, and the S7 part represents the Dirac electron and red,green, and blue up and down quarks.

The RP 1 part is represented by [0, 1) in our parameterization (or [0, π)on the unit circle in the complex plane in a more conventional one), and theS7 part can be represented by the unit sphere S7 in the octonions O.

Also, mathematically, we can regard

S7 ×RP 1 = (S7 ×RP 1)† (5)

Therefore, the creator-annihilator part of the nilpotent Heisenberg 3× 3matrix can be represented as:

0 S7 ×RP 1 0

0 0 S7 ×RP 1

0 0 0

(6)

5

What about the β part?

β is given by the commutator

β = [S7 ×RP 1, S7 ×RP 1] (7)

Since RP 1 is only the interval [0, 1) in our parameterization (or [0, π) onthe unit circle in the complex plane in a more conventional one), it is equiv-alent to a real number and can therefore be absorbed into the real R scalarfield of the 3× 3 matrices.It commutes with everything and produces no gauge bosons by its commu-tators.

From a physical point of view, we can say that RP 1 represents the neu-trino, which has no charge and therefore does not interact with or produceany gauge bosons by commutation.

Whichever point of view you prefer, the result is that the full 3 × 3nilpotent Heisenberg matrix looks like:

0 S7 β

0 0 S7

0 0 0

(8)

Therefore, β is given by

β = [S7, S7] (9)

Unlike the parallelizable spheres S1 and S3 of the associative algebrasC and Q, the 7-spehre S7 of the nonassociative octonions O does not closeunder commutator and does not form a Lie algebra.

To deal with the situation, we need to use Martin Cederwall’s octonionX-product [2] and Geoffrey Dixon’s XY-product [7].

6

Martin Cederwall and his coworkers [2] have shown that [S7, S7] doesform an algebra, but not a Lie algegbra:

Consider a basis {eiX} of the tangent space of S7 at the point X on S7.Following Cederwall and Preitschopf [2], we have

[eiX, ejX ] = 2Tijk(X)ekX (10)

Due to the nonassociativity of the octonions, the ”structure constants”Tijk(X) are not constant, but vary with the point X on S7, producing torsion.

Effectively, each point of S7 has its own X-product algebra.

The X-product algebra takes care of the case of [eiX, ejX] where both ofthe elements are in the tangent space of the same point X of S7, but sincedifferent points have really different tangent spaces due to nonassociativityof the octonions, it does not take care of the case of [eiX, ejY ] where eiX isan element of the tangent space at X and eiY is an element of the tangentspace at Y .

To take care of this case, we must use Geoffrey Dixon’s XY-product and”expand” [S7, S7] from S7 to at least two copies of S7 (one for the commutoralgebra at each of the points of the other one). That is, if 1 denotes afibration ”product”:

[S7, S7] ⊃ S71 S7 (11)

We are still not quite through, because even though we have used theXY-product to take care of the case of [eiX, ejY ] where eiX is an element ofthe tangent space at X and eiY is an element of the tangent space at Y , wehave not taken into account that the octonion basis for the tangent spce atat X may be significantly different from the octonion basis for the tangentspace at Y .

7

The extra structure that must be ”added” to S71 S7 to ”transform”

the tangent space at X into the tangent space at Y is the automorphismgroup G2 of the octonions. Unlike the cases of the associative algebras, theaction the automorphism group cannot be absorbed into the products wehave already used. So, we see that the Lie algebra of [S7, S7] is

[S7, S7] = S71 S7

1 G2 = Spin(8) (12)

The fibrations represented by the 1 are:

Spin(7)→ Spin(8)→ S7 (13)

andG2 → Spin(7)→ S7 (14)

Now, our octonionic version of the nilpotent Heisenberg algebra lookslike:

0 S7 Spin(8)

0 0 S7

0 0 0

(15)

Here, Spin(8) is the 28-dimensional adjoint representation of Spin(8). Its28 infinitesimal generators represent 28 gauge bosons acting on the fermionsthat we have created, all as in the D4 −D5 − E6 model.

The action of the Spin(8) gauge bosons takes place within the arena of the8-dimensional vector representation of Spin(8), again as in the D4−D5−E6

model.

We now have the picture of fermion creators and annihilators forminggauge bosons, and all of them interacting in accord with the D4 −D5 − E6

model.

8

However, what about spacetime?

Since by triality (Porteous [11] describes triality) the vector representationof Spin(8) is isomorphic to each of the half-spinor representations that weuse for fermion creators and annihilators, we can form a vector representationversion of the octonionic nilpotent Heisenberg algebra.

0 S7 S7

0 0 S7

0 0 0

(16)

If we put back explicitly the factors of RP1 that we had merged into thereal scalar field for ease of calculation of the S7 commutators, we get:

0 S7 ×RP 1 S7 ×RP 1

0 0 S7 ×RP 1

0 0 0

(17)

The vector Spin(8) spacetime part is

0 0 S7 ×RP 1

0 0 0

0 0 0

(18)

It is represented in the D4 −D5 − E6 model by the Shilov boundary ofthe bounded complex homogeneous domain corresponding to the Hermitiansymmetric space D5/(D4 × U(1)).The Shilov boundary is S7 ×RP 1. The RP 1 part represents the time axis,and the S7 part represents a 7-dimensional space.

9

NOW, we have reproduced the structure of the D4 −D5 − E6 model bystarting from octonion fermion creators and annihilators.

We can therefore incorporate herein by reference all the phenomenologicalresults of the D4 −D5 − E6 model as described in hep-ph/9501252 [13].

3 Complexified Octonions.

Recall that the octonion fermion creators and annihilators are of the form

0 S7 ×RP 1 0

0 0 S7 ×RP 1

0 0 0

(19)

and that both of the entries S7 × RP 1 taken together are representedin the D4 − D5 − E6 model by the Shilov boundary of the bounded com-plex homogeneous domain corresponding to the Hermitian symmetric spaceE6/(D5 × U(1)).

Also recall that the vector Spin(8) spacetime part

0 0 S7 ×RP 1

0 0 0

0 0 0

(20)

is also represented in the D4 −D5−E6 model by a Shilov boundary of abounded complex homogeneous domain. This entry S7 ×RP 1 correspondsto the Hermitian symmetric space D5/(D4 × U(1)).

10

What if, instead of representing the 3 × 3 nilpotent Heisenberg matrixstructure by Shilov boundaries, we represent them by the linearized tangentspaces of the corresponding Hermitian symmetric spaces?

Then we would have:

0 C⊗O C⊗O

0 0 C⊗O

0 0 0

(21)

Sarah Flynn uses such 3× 3 matrix structures in her work [8].Note that complexified octonions C⊗O are not a division algebra.That is because signature is indistinguishable in complex spaces.Therefore, both the octonions and the split octonions are subspaces of C⊗O.Since the split octonions contain nonzero null vectors, the complexified octo-nions C⊗O may be a normed algebra, but they are not a division algebra.The only complex division algebra is the complex numbers C themselves.

4 Dimensional Reduction.

Now, going back to the Shilov boundary uncomplexified representations, re-call that the vector Spin(8) spacetime is represented by

0 0 S7 ×RP 1

0 0 0

0 0 0

(22)

Here, the spacetime of the vector representation of Spin(8) is S7×RP 1,which can be represented by the octonions if RP 1 is the real axis and S7 isthe imaginary octonions.

11

How do we move a fermion created at one point to another point?

If we move a particle along a lightcone path, how do we tell how ”far” ithas gone?

Following the approach of John Caputlu-Wilson [1], we should measurehow much its propagator phase has advanced.

Since the phase advance may be greater than 2π, the propagator phaseshould take values, not on the unit circle, but on the infinite helical multi-valued covering space of the unit circle.

Recognizing that it may be difficult to do an experiment that will distin-guish phases θ greater than 2π from phases θ−2π, we will look at very shortpaths such that the phase advance along the path is much less than 2π.

Now that we have a way to tell how ”long” is a lightcone path segment,we can look at some paths.Consider the following two lightcone paths P1 and P2, each beginning at Xand ending at Y and each made up of two ”short” lightcone segments:

��@I@I�� P2P1

Y

X

Since the octonion spacetime S7 ×RP 1 is nonassociative, it has(as Martin Cederwall and his coworkers have shown [2])torsion.

Since it has torsion, the end-point Y may not be well-defined, and wemay have the diagram:

��@IAAK��� P2P1

Y

X

Since we want paths and lightcones to be consistently defined in the

12

Minkowski vacuum spacetime(before gravity has acted to effectively distort spacetime)we must modify our octonionic spacetime so that it is torsion-free at theMinkowski vacuum level.

How do we get rid of the torsion?

We must get rid of the nonassociativity.

To do that, reduce the octonionic spacetime S7 × RP 1 to its maximalassociative subspace.

How do we determine the maximal associative subspace of the octonionicspacetime S7 ×RP 1?

Following Reese Harvey [9], define the associative 3-form φ(x, y, z) forx, y, x ∈ S7 by:

φ(x, y, z) =< x, yz > (23)

where < x, yz > is the octonion inner product Re(xyz) .

The associative form φ(x, y, z) is a calibration that defines an associativesubmanifold of S7.

When combined with the real axis part RP 1 of octonion spacetime, theassociative submanifold of S7 gives us a 4-dimensional quaternionic associa-tive spacetime submanifold of the type S3 ×RP 1.

4-DIMENSIONAL QUATERNIONIC SPACETIME S3 ×RP 1

IS THE ASSOCIATIVE PHYSICAL SPACETIME.

This structure is the same as that of the D4−D5−E6 model. A detaileddescription of how dimensional reduction works in the D4 −D5 −E6 model,including its effects on fermions and guage bosons, is given in hep-ph/9501252[13].

13

5 Spacetime and Internal Symmetries.

Much of the material in this section is taken from the book of Reese Harvey[9]. To the extent that this section is good, he deserves credit. To the extentthat this section is wrong or bad, it is not his fault that I made mistakesusing his book.

The 4-dimensional associative physical spacetime is determined by theassociative 3-form φ(x, y, z) on S7 defined in the previous section.

What happens to the rest of the original 8-dimensional spacetime?

It is the orthogonal 4-dimensional space determined by the coassociative4-form ψ(x, y, z, w) on S7 defined for x, y, z, w in S7 as

ψ(x, y, z, w) = (1/2)(x, y(zw)− w(zy)) (24)

That means that the original 8-dimensional spacetime S7 × RP 1 is de-composed into an associative physical spacetime Φ = S3×RP 1 and a coasso-ciative internal space Ψ determined by the coassociative 4-form ψ(x, y, z, w)on S7.

If the associative physical spacetime Φ is taken to be the real part andthe coassociative internal space Ψ is taken to be the imaginary part of acomplex space Φ + iΨ, then the full spacetime is transformed from a real8-dimensional space, locally R8, to a complex 4-dimensional space, locallyC4.

The gauge group Spin(8) acting locally on R8 is then reduced to U(4)acting locally on C4.

5.1 Spacetime, Gravity, and Phase

We now have the gauge group U(4) acting on the associative physical space-time Φ = Re(C4).

14

Since U(4) = Spin(6) × U(1), and Spin(6) is the compact version ofthe 15-dimensional conformal group, we can now build a model of gravityby gauging the conformal group Spin(6) and use the U(1) for the phase ofpropagators in the associative physical spacetime.

Note that only the 10-dimensional de Sitter gauge group Spin(5) sub-group of the Spin(6) conformal group is used to build gravity.

The other 5 degrees of freedom are 4 special conformal transformationsand 1 scale dilatation. The 4 special conformal transformations are gauge-fixed to pick the SU(2) symmetry-breaking direction of the Higgs mechanism,and the scale dilatation is gauge-fixed to set the Higgs mass scale.

For details, see hep-ph/9501252 [13] and [12].

5.2 Internal Space and Symmetries.

Now we have:

associative physical spacetime Φ = S3 ×RP 1 = Re(C4);

gravity from the conformal group Spin(6);

Higgs symmetry breaking and mass scale from conformal Spin(6); and

propagator U(1) phase.

We have not yet built anything from:

the coassociative imaginary space Ψ = Im(C4) ; or

the part of the gauge group Spin(8) that is in the 12-dimensional cosetspace Spin(8)/U(4) .

Let the coassociatve imaginary space Ψ = Im(C4) be the internal sym-metry space on which the internal gauge groups act transitively.

That means that Ψ = Im(C4) plays a role similar to the internal sym-

15

metry spheres of Kaluza-Klein models.

Let the part of the gauge group Spin(8) that is in the 12-dimensionalcoset space Spin(8)/U(4) be the internal symmetry gauge groups.

A problem is presented here:The coset space is just a coset space, with no group action.How does it represent internal symmetry gauge groups?

The 12-dimensional coset space Spin(8)/U(4) is the set of oriented com-plex structures Cpx+(4) on R8, and is also the Grassmannian GR(2,O).

Each element of the Grassmannian GR(2,O) can be represented by asimple unit vector in

∧2 O.

Each simple unit vector in∧2 O determines a reflection, and all those

reflections generate the group Spin(8).

Geometrically, what we have is that the12-dimensional coset space Spin(8)/U(4) can be representedby 12 ”positive” root vectors in the 4-dimensional rootvector space of the D4 Lie algebra of Spin(8),

while the 16-dimensional U(4) subgroup of Spin(8) can be representedby the 12 ”negative” root vectors plus the 4-dimensional Cartan subalgebraof the 4-dimensional root vector space of the D4 Lie algebra of Spin(8).

Using quaternionic coordinates for the root vector space,

{±1,±i,±j,±k, (±1± i± j ± k}

are the 24 root vectors, and the 12-dimensional coset space Spin(8)/U(4)can be represented by the 12 root vectors

{+1,+i,+j,+k, (+1± i± j ± k)/2}

16

What internal symmetry gauge groups do the 12 coset space Spin(8)/U(4)generators of Spin(8) form?

Since the 12 coset space Spin(8)/U(4) generators can be represented bythe quaternions

{+1,+i,+j,+k, (+1± i± j ± k)/2}

and since they do not together form a simple Lie group, consider what carte-sian product of simple Lie groups might be formed.

The 8 quaternions{(+1± i± j ± k)/2}

should form the Lie group SU(3), with, for example, (+1 + i+ j + k)/2 and(+1− i− j − k)/2 as its Cartan sualgebra.

The 3 quaternions{+i,+j,+k}

should form the Lie group SU(2), with, for example, +j as its Cartansualgebra.

The remaining quaternion{+1}

should form the Lie group U(1), which is Abelian and equal to its Cartansualgebra.

Therefore, in this model the 12-dimensional coset space Spin(8)/U(4)represents the internal symmetry group of the Standard Model

SU(3)× SU(2) × U(1)

.

The 4-dimensional internal symmetry space Ψ is the representation spaceon which each of the internal symmetry groups acts transitively.

The de Sitter Spin(5) of the U(4) = Spin(6)×U(1) also acts transitivelyon the imaginary internal symmetry space Ψ.

17

Each of the 4 groups Spin(5), SU(3), SU(2), U(1) act transitively on the4-dimensional internal symmetry space Ψ with its own measure.

Effectively, each measure is determined by the way in which the gaugebosons of each of the 4 forces ”see” the 4-dimensional internal symmetryspace Ψ.

The way each ”sees” the space is determined by the geometry of the 4-dimensional symmetric space Ψforce on which each force acts transitively:

Gauge Group Symmetric Space Ψforce

Spin(5) Spin(5)Spin(4)

S4

SU(3) SU(3)SU(2)×U(1)

CP 2

SU(2) SU(2)U(1)

S2 × S2

U(1) U(1) S1 × S1 × S1 × S1

(25)

More about this is in

WWW URL http://www.gatech.edu/tsmith/See.html [12].

The ratios of the respective measures are used to calculate the relativeforce strength constants in this D4−D5−E6 model. For detailed calculationsof force strengths (and also particle masses and K-M parameters), see hep-ph/9501252 [13] and [12].

Not only does the 10-dimensional de SitterSpin(5) of the U(4) = Spin(6) × U(1) act onthe imaginary internal symmetry space, butthe (4+1)-dimensional conformal Higgs mechanism acts onthe internal symmetry space to give mass to

18

the SU(2) weak bosons, andthe U(1) propagator phase acts to give phases to the gauge bosons.

NOW, we have constructed the D4 − D5 − E6 model that includes theStandard Model plus Gravity, all from the beginning point of fermion creatorsand annihilators.

This construction of the model uses a continuous spacetime.A future paper will deal with a discrete HyperDiamond lattice generalizedFeynman checkerboard version of the model.

19

References

[1] J. Caputlu-Wilson, gt6465a@prism.gatech.edu, is working on the prop-erties of the phase of propagators.

[2] M. Cederwall and C. Preitschopf, S7 and S7, preprint Goteborg-ITP-93-34, hep-th/9309030.

[3] G. Dixon, Division algebras: octonions, quaternions, complex numbers,and the algebraic design of physics, Kluwer (1994).

[4] G. Dixon, Octonion X-product Orbits, hep-th/9410202.

[5] G. Dixon,Octonion X-product andOctonion E8 Lattices, hep-th/9411063.

[6] G. Dixon, Octonions: E8 Lattice to Λ16, hep-th/9501007.

[7] G. Dixon, Octonion XY-Product, hep-th/9503053.

[8] S. Flynn, gt6465a@prism.gatech.edu, is working on 3× 3 matrix modelsusing octonions.

[9] R. Harvey, Spinors and Calibrations, Academic (1990).

[10] S. Helgason, Geometrical Analysis on Symmetric Spaces, AmericanMathematical Society (1994).

[11] I. Porteous, Topological Geometry, 2nd ed, Cambridge (1981).

[12] F. Smith, WWW URL http://www.gatech.edu/tsmith/home.html.

[13] F. Smith, Gravity and the Standard Model with 130 GeV Truth Quarkfrom D4 − D5 − E6 Model using 3 × 3 Octonion Matrices, preprint:THEP-95-1; hep-ph/9501252.

20